the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 13 Mar 2023
A linear assessment of waveguidability for barotropic Rossby waves in different large-scale flow configurations
Abstract. Topographically forced Rossby waves shape the upper-level waveguide over the midlatitudes, affecting the propagation of transient waves therein, and have been linked to multiple surface extremes. The complex interplay between the forcing and the background flow in shaping the Rossby wave response still needs to be elucidated in a variety of configurations. We propose here an analytical solution of the linearized barotropic vorticity equation to obtain the stationary forced Rossby wave resulting from arbitrary combinations of forcing and background zonal wind. While the onset of barotropic instability might hinder the applicability of the linear framework, we show that the nonlinear wave response can still be retrieved qualitatively from the linearized solution. Examples using single- and double-jet configurations are discussed to illustrate the method and study how the background flow can act as a waveguide for Rossby waves.
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RC1: 'Comment on egusphere-2023-316', Anonymous Referee #1, 24 Apr 2023
Summary
This manuscript offers a new method (to the best of my knowledge) for solving the barotropic vorticity equation linearized about a background solution, with a topographic forcing term. This method is based on expressing the solution as a superposition of Chebyshev polynomials and expressing the derivative operators using the Chebyshev basis. The authors argue that this method is computationally efficient, relative to numerical integrations, when examining the wave response to changes in the forcing or in the background flow. This method also allows for a calculation of the eigenvalues of the problem, given the background flow and regardless of the forcing, which allows for an examination of the stability of the flow to linear perturbations. The authors compare the solutions calculated using this method with numerical solutions of the fully nonlinear barotropic vorticity equation, with specific background flow configurations. The solutions are found to be quite similar, confirming the validity of the method proposed. The authors discuss some implications of this methods for estimating the waveguidability of different zonal flows.
I think the method and results presented in this manuscript are interesting and significant, and could be useful for future studies. I have major comments regarding the presentation of the manuscript. I personally had to read through the manuscript carefully twice before I had a sense of what it is about. I think the manuscript can be improved by making some modifications in the text, to make the ideas clearer.
Major comments
- In my opinion, the introduction, the abstract and perhaps even the title do not express clearly what the paper is about. They give the impression that the paper is more about the physics of waveguides, whereas the main contribution of the paper, as I see it, is the computational method. The single and double jet cases shown in the paper are used as test cases for the computational method, rather than going deep into the dynamics of waveguides. In the last paragraph of the paper the authors write that “This study should be regarded as an introduction and explanation of the techniques, but possible applications of this approach could include systematic waveguidability assessments for different forcings and background zonal wind profiles.” I think this sentence should appear in the introduction, and the abstract should emphasize the main point of the paper.
- Perhaps the authors could add some more motivation for specific choices in the derivation of the mathematical model, that could add to the clarity of the paper. One choice that wasn’t immediately clear to me was including a damping term in equation (1). It wasn’t clear what this damping term represents physically. It was also not clear to me what the motivation was for looking at the stationary solution in equation (13). Only after I finished reading it became clearer. I think that in section 2 it could help to explain better what the model is supposed to represent, before the derivation of the equations.
- The authors present stationary solutions and time-dependent solutions, but it is not mentioned explicitly at which section each type of solution is examined, what each type of solution represents physically, what is the motivation for looking at each type of solution and what are the different methods for solving for a stationary or time-dependent solution. Each of these are explained somewhere, often after presenting the results, but it is not explained in an organized manner.
- Section 4.3 analyzes the stability of the problem for different parameters of the jet profile. I was missing a discussion on the connection of this instability to linear barotropic instability, in the sense of the necessary conditions for instability including a change of sign of the PV gradient. Some more physical context would be useful.
- The analysis of the double jet case shown is much shorter than that of the single jet case and does not include a calculation of waveguidability. Based on the last sentence of the abstract (“Examples using single- and double-jet configurations are discussed to illustrate the method and study how the background flow can act as a waveguide for Rossby waves”) I was expecting a comparison between the waveguidability of the two cases. If the authors choose not to include a calculation of waveguidability for the double jet case, they should otherwise motivate the choice for the analysis that is presented.
Minor comments
- The term “analytical solution” used in the abstract and introduction was confusing for me. I expected to see a solution expressed as a mathematical function. It is true that the Chebyshev polynomials are analytical functions and in that sense the solution is analytical, but eventually there is a numerical calculation that leads to the solution, so perhaps a different terminology would describe the method more clearly. As I see it, the main difference between what is called a “numerical solution” and “analytical solution” by the authors is that the former is a time-integrated solution and the latter is an eigenvalue problem.
- Lines 181-182: In what sense is it counterintuitive that the nonlinear solution is more dissipative?
- Perhaps the appendix can include some more details of the computational method, such as the matrices D(1) and D(2), and how the time-dependent solution is calculated.
Citation: https://doi.org/10.5194/egusphere-2023-316-RC1 -
AC1: 'Reply on RC1', Antonio Segalini, 23 May 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-316/egusphere-2023-316-AC1-supplement.pdf
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RC2: 'Comment on egusphere-2023-316', Anonymous Referee #2, 25 Apr 2023
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AC2: 'Reply on RC2', Antonio Segalini, 23 May 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-316/egusphere-2023-316-AC2-supplement.pdf
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AC2: 'Reply on RC2', Antonio Segalini, 23 May 2023
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